Development of a process simulator using object oriented programm.pdf

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Development of a process simulator using objectoriented programming: Numerical procedures and

convergence studiesKheng Hock LauIowa State University

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Order

Number 9234828

Development of a process simulator using object-oriented

programming: Numerical procedures and convergence studies

Lau, Kheng Hock, Ph.D.

Iowa State

University,

1992

U M I

300

N.

Zecb

Rd.

Ann

Arbor, MI

48106


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6/148

Development

of

a

process simulator using

object

oriented

programming:

Numerical procedures and convergence studies

b y

Kheng Ho ck Lau

A Dissertation Submitted to

the

Graduate

Faculty

in Partial

Fulf i l lment

of the

Requirements

for

the

Degree

of

DOCTOR

OF

PHILOSOPHY

Major: Chemical Engineering

App^ed:

In

Charge of Maior

W or k

For the Major

Dt^artment

F or the

Graduate

Col lege

I owa

State

U nive r s i t y

A m e s ,

Iowa

1 9 9 2


7/148

i i

TABLE OF

CONTENTS

ACKNOWLEDGMENTS viii

CHAPTER

1.

INTRODUCTION 1

CHAPTER

2.

LITERATURE

REVIEW 5

Process

Simulation

5

Sequential

modular simulators 5

Equation-based simulators 8

Numerical Methods 9

Tearing Algor i t hms il

Densities f rom

Equations

of

State

14

Object

Oriented

Programming 16

CHAPTER

3. NUMERICAL

PROCEDURES

19

Numerical

Methods 19

Newton's method 19

M o d i f i e d

Powell ' s

dog leg

method

2 1

Tearing

Algorithms

40

Partitioning

algorithm 42

Physical

Property

System 59

Top l i s s ' s

algorithm

61


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Mathias's algorithm 6 2

Property

method system 71

CHAPTER 4. CONVERGENCE STUDIES 8 1

Sequential Modular Simulators

82

Processes simulated 83

Results 87

Equation-Based Simulators 9 3

CHAPTER

5. CONCLUSIONS

1 0 1

CHAPTER 6. RECOMMENDATIONS 1 0 4

BIBLIOGRAPHY 1 0 6

APPENDIX A. TEST FLOWSHEETS

1 1 2

APPENDIX B. FULL-MATRIX TEST PROBLEMS 1 1 8

APPENDIX C.

SAMPLE

C++

PROGRAMS

1 2 6


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iv

LIST OF TABLES

Table

3.1: Test

problems

for ful l matrix

modi f i ed Powel l ' s

method

... 24

Table 3.2: Number of iterations for full matrix modi f i ed P owel l ' s method

in test

runs

27

Table

3.3:

Test problems for sparse matrix

modif ied Powel l ' s method .

. 33

Table

3.4:

Number of

iterations for sparse matrix modi f i ed

Powel l ' s

method

in

test

runs 3 4

Table

3.-5:

Class ical flowsheets

55

Table 3 .6 : Legend for tearing algorithms 55

Table 3.7:

Number of

tear streams selected

56

Table 3 .8 : Averaged execution time (ms) 57

Table

4.1:

Tear sets u sed in simulations 8 6

Table B.l: Parameter set 123


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V

LIST

OF FIGURES

Figure

1.1 : Elements of an Object Oriented Process

Simulation

Environ

ment

4

Figure

2.1:

Sample

flowsheet

with

a r ecycle l oop 7

Figure 3.1: A sample occurence matrix

2 8

Figure 3.2 : A C++

program demonstrating

the

use

of nonlinear equation

so lver

37

Figure 3.3 : C++ code for Broyden's

update

3 8

Figure

3.4:

Simpl i f i ed

Barkley-Motard graph

44

Figure 3.5 :

Simpl i f i ed

Christensen-Rudd graph

II

48

Figure 3.6 :

Adjacency Matrix

for graph

in Fig. 3.5

48

Figure 3.7 :

Tie

reso lu t ion scheme 50

Figure 3.8 : Adjacency

Matrix

for Figure 3.4 51

Figure 3.9 : Three typical curves for

P( /9 )

versus

density

63

Figure 3.10:

Three typical c u r ve s for P p [ p ) ve r sus

density

6 4

Figure

3.11: L iqu id

densities

for

an

equal-molar

mixture

of

ethane

and

n -

heptane with

SRK equation

of state

6 9


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vi

Figure 3 .12 : Vap o r densities for an equal-molar mixture

of

ethane and n -

heptane with SRK equation of state

70

Figure 3.13: Fugacity

coeff icients of an equal-molar mixture

of

ethane and

n-heptane with SRK equation

of

state 72

Figure 3.14: Fugacity coeff icients of an equal-molar mixture

of ethane and

n-heptane

with SRK

equation of state 73

Figure 3.1.5:

Enthalpies

of an

equal-molar mixture

of ethane

and n-heptane

with SRK equation of state

74

Figure

3 .16 :

Enthalpies

of

an

equal-molar

mixture

of

ethane and n-heptane

with

SRK

equation of state 75

Figure 3.17: A C-f-f sample program

us ing

phys ica l property system ... 80

Figure

4.1 : Insertion of

inner loop

84

Figure 4.2 : Cavett problem

86

Figure

4.3 : Recompression f lowshee t 8 7

Figure

4.4: A

s ing le - loop flowsheet 88

Figure

4.5 : Iterations vs tear set for

Cavett

problem

(Broyden's

Method) 89

Figure

4.6 :

Iterations vs

tear

sets for Cavett problem

(direct substitution) 9 0

Figure

4.7:

Iterations v s

tear

sets

for

Cavett

problem (SQP) 9 1

Figure 4.8 : Iterations v s

tear sets

fo r r e c omp r e s s ion f l owshe e t (Broyden's

Method) 92

Figure

4.9 :

Iterations

v s

tear

sets

for

r e c omp r e s s ion f l owshe e t

(direct sub

stitution)

9 3

Figure

4.10: Iterations vs

tear

sets for r e c omp r e s s ion flowsheet

(SQP)

. .

9 4


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v

Figure 4.11; Iterations v s tear sets for s ing le - loop

flowsheet

(Broyden's

method) 9 . 5

Figure A.l:

Rubin

graph

112

Figure A.2:

Rubin and

Hutchison

graph 112

Figure A.3: Christensen and Rudd graph 112

Figure A.4 : Upadhye and Grens graph 113

Figure

A.5 : Christensen

and Rudd

graph I 113

Figure A.6 :

Christensen

and

Rudd graph

II 113

Figure

A.7 : Sargent and

Westerberg

graph 114

Figure A.8:

Shannon graph (Su lfu r ic

ac id ) 114

Figure

A.9; Gundersen graph

(Heavy

water) 115

Figure

A.10:

Pho and Lapidus

graph

115

Figure

A.11:

Barkley

and Motard graph 1 1 6

Figure

A.12: Jain

and Eakman

graph

(HF-alkylation) 1 1 6

Figure A.13: Jain and Eakman graph

(Vegetable

oi l ) 117

Figure C.l:

A

s ing le - loop flowsheet

127


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viii

ACKNOWLEDGMENTS

I would l ike to express my sincere gratitude to Pro fe s so r Dean L. Ulr ichson

for

his

guidance and

advice throughout the course of this research. Many

thanks to

Professor

Les

L. Mil le r for sharing h i s ve ry

interesting

perspective on databases with

our research group and attending many of our group meetings. Many important ideas

were

ident i f ied

in

the

discuss ions .

The financial assistance I

received

from the

Department of

Chemical Enginee r

ing at

Iowa

State

whi l e work ing on this research is deeply appreciated. Thanks to

Pro fe s so r Richard S. Seagrave

for

kindly lending me

one

of his group's workstations.

Special

thanks

to my fel low co -worke r

Mr. Gadiraju

Varma from

w h o m I

learned

so many

programming

t r icks .

Thanks for bringing

me into

the wor ld

of object

oriented

programming.

My parents shall be complimented fo r a l l owing

me

to explore further education

adventures in America.

The initial

opportunity that I was g iven has made

a subtle

impact

in my

l ife

and

brought me into the

boundary of science and engineer ing .

Finally, I am grateful

to my f iancee ,

Masami

lida

for

her patience,

understanding

and support.


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1

CHAPTER 1.

INTRODUCTION

Applications of

computers

in

chemical

engineering

began approximately thirty-

f iv e yea r s

ago [26 ,

2 7 , 48 ,

49] .

Initial work

was

mainly concentrated

on

dev e lop ing

application

specif ic

computer programs.

There

was

little

thought

giv en

to the

concept

of a

proces s

simulator. Howev er , the initial

efi 'ort

produced many

re l iab le

algorithms

s pec ia l ly

developed

for

s o lv ing

process

simulation problems. For example,

several

al

gorithms w e r e d e v e l o p e d to s o lv e the

resulting energy and

material

balance

equations

and the

gov ern ing

equations that arise

from

phase equilibrium thermodynamics for

multi-stage operations.

B y

the

end

of

1960 s ,

it was evident that if al l the prev iou s ly

dev e loped

programs

were

incorporated

into

a

single system

that

cou ld

perform

al l

the

neces s a ry

calculations for an

entire or a

section of a

f lows h ee t , then proces s eng i

nee r s

would have a powerfu l tool

that

could

expand

the

boundary

of the simulation

problems that

cou ld

be solved

[69] .

Consequently,

proces s

simulators

were

developed.

With today's

computing t ech no log ies , p roces s

simulation

is

performed routinely

in the des ign of new proces s es and studies

of

current processes related to

p lan t

oper

ations,

optimization, proces s

control, and operability. Such a

wide

array

of tasks cer

tainly requires a

systematic approach

to organize

al l information generated

through

out

the

course of a

project. In addition, the

project

work

may

involve d i f fe ren t

application software

pack ages that

perform a

range

of

computations f r o m

simple

unit


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2

simulation

to

rigorous

p r oc es s

simulation. O f course,

these

sof tware packages

may

exchange data

during the

process calculations. Hence,

the

creation of a common

framework for these application software packages will certainly

improve

engineering

productivity and

enhance the

tools

available

for process

studies.

This concept has

been ca l led process

integration. Initial

work in this

area

i nv o lv ed

the

extension of

current p r oc es s f lowsheet ing pac kages

and

the development of eng i nee r i ng database

management systems. Many obstacles were encountered in integrating these appli

cation

programs

because

most

of them

w er e de s i gned with no

intention of

being

embedded into another

system. R ec ogn i z i ng

the

need

for establishing data trans

fer

standards, AIChE has

already

formed

the

Process

Data

Exchange Institute

to

develop such standards for process integration [69] .

Object oriented programming

(OOP) has

been recognized by severa l r esearchers

as

having

the potential

for

achieving process integration in deve l op i ng engineering

softwares [61] . In

OOP,

data types

are

modeled in a natural fashion for

the

application

of interest. Fina l ly , programs

are

composed from

the

ident i f i ed data types obtained

from

data modeling o r data abstraction. It is

c lear

that OOP gives more importance

to

data as opposed

to operations

that

u se the

data

to

perform

the

des i red

function.

An object is a

data

type

with

its respective def ined operations. Si nc e OOP is

still

in

its

infancy in c hem i c a l engineering

application,

work in

exploring

h ow

w e

can use

OOP

in engineering software development

is

ve r y much

under

active research.

The current work involved the development

of

a

process

simulation

environment

us i ng object oriented programming.

Our

view of an integrated chemica l p rocess sim

ulation environment is shown in Figure 1.1. The pac kage c ons i s t s of a sequential

modular

simulator,

an

equation-based simulator,

a phys i c a l

property system,

a phys


16/148


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4

Physical

P r op e r ty

Sys tem

Physical

Quant i ty

S ys t e m

Equa t ion

Based

Simulator

Databank

Sequent ia l

Mo du l a r

S imula to r

C

+ +

Figure

1.1: Elements

of

an Object

Oriented

Process Simulation Environment


18/148

5

CHAPTER 2.

LITERATURE

REVIEW

Process

Simulation

Simulation

of

c he mic a l

plants

is very common with

today's

advanced computing

t echnologies .

For

steady

state material

and energy

balance

computations, a c he mic a l

plant

can be described

b y

a

set

of nonlinear algebraic equations.

Since

the advent

of the f i rs t

simulator

in c he mic a l engineering, there have been numerous simulators

developed in both

academic

and industrial environments. For example, CHESS [68],

FLOWTRAN [48],

ASPEN [1],

PROCESS [68] , FLOWSIM

[55]

and SPEEDUP [43].

G ive n a

f lowshee t ,

all these

simulators can

so lve the resulting set of nonlinear a lge b r a i c

equations. There are three primary achitectures

fo r c he mic a l p r oc e s s

simulators

in

use today. They are sequential modular simulators, equation-based simulators and

simultaneous

modular

simulators.

This work concerns only sequential

modular

and

equation-based simulators.

Sequential

modular

simulators

This type

of

simulator

is ve r y

common in

the

industry.

Biegler

[4]

recent ly

performed a survey of

current

commercial simulators and found that se ve n ty p e r c e n t

of

the

simulators are sequential modular simulators. A s the name implies,

a

sequential

modular simulator performs calculations

sequentially

in

the

direction of material


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6

flow.

This

imp l i e s

that

b efore

the

computations

for one particular process unit

in

a

f lows h ee t are

performed, all

input streams

to

that

unit must

be known. With

the

presence of recyc le streams,

the

output of

one

downstream proces s

unit is the

input

of o n e upstream proces s unit.

Consequently,

the calculations are iterative because

some stream's state and component

flowrates

must

be gu es s ed and

corrected

till

they conv erge

within a

prespeci f ied tolerance.

The

gu es s ed streams are

commonly

r e fe r r ed to

as

tear streams. It is

obvious

that gu es s ed streams

h a v e

to be

selected

b efore

simulation

computations can

begin if the f lowshee t

of

interest

contains r ecyc le

streams. A

sample flowsheet

is shown in Figure

2.1 .

The

se lec ted tear stream is

Recyp. The

resulting set of

equations can

be arranged as

X

=

Y (2.1)

or

X - Y

= 0 ( 2 . 2 )

X is the

gu es s ed

stream vector containing component

flowrates,

enthalpy and

pres

sure. Y is the computed stream

vector consisting

of

elements

s imi lar

to

X. Equa

tion 2.1 is usually

s o lv ed

using Wegstein's

method

[2] or direct substitution. In

order

to use Newton's method or Broyden's

method [6 ] ,

the set

of

equations is expressed

in

the form

of

equation 2.2 .

One

obvious characteristic

of this

set

of

equations is that

the

equations can

not

be written

explicitly

in terms of the

unknowns. Consequently,

in order to evaluate equation 2.1 or

2.2,

the

f lows h ee t

has

to be evaluated. This can

be expensive for methods

that

require

derivative

information.

S ev era l

algorithms

h av e

been

developed

to

identify

the optimal tear streams

b as ed on the t opo logy of flowsheets.

All these algorithms ignore

the

existence

of c o n -


20/148

I

Feedf Vap

Liq

ecy

Recyp

Prod

Y

=

Spit

Flash

Pump

Mixer

Figure 2.1: Sample f l owshee t with a recycle

loop

straints when gues sed streams are selected

as

l ong

as

the f inal

computation sequence

is consistent.

A

constraint is

a des ign spec i f i ca t ion thru

which

w e w i sh to

control

some

output

of a p roces s unit. When constrain ts are imposed, extra inner l oops have to be

converged in addition

to

the

outer

iterative

computation

described above. Because of

the architecture of sequential modular

simulators,

constraints nearly a lways

compl i

cate

the

conve rgence

of

iterated variables. In

general, f l owshee t s almost

a lways

have

recycle

streams and

constraints.

This

kind of

simulation

is termed des ign simulation.

On

the

contrary,

if

there

is no

constraint,

the simulation is cal led performance simu

lation. Since

the convergence behavior

of sequential modular simulators

is generally

more complicated when des ign simulation is performed,

there

is a

need to

investigate

the

effect

of

constraints on the

performance of

sequential modular simulators.


21/148


22/148

9

that

is robust

enough to

satisfy the

needs

of proces s simulation. One famous

com

mercial equation-based simulator is SPEEDUP

[43] ,

deve l oped at

Imperial Col lege ,

London. Prototype equation-based

simulators

have

a l s o b een

developed

at

several

institutions. For instance, FLOWSIM

(University

of Connecticut) [-55], ASCEND

(Carnegie M e l l o n )

[40] ,

and

SEQUEL (University

of I l l inois) [60] .

We developed a

prototype

ec ju a t ion -b as ed

simulator

to

illustrate the applicability

of object oriented

programming i n p roces s

simulation.

Then

the performance

of

this

equation-based simulator

was

evaluated.

Numerical Methods

For both sequential modular and equation-based simulators,

a

nonlinear ecpia t ion

solver i s needed. There have been

two

competitive nonlinear equation

solvers

in

the literature. One is the

class ic

quasi-Newton method which is mostly k n o w n as

Broyden's method.

There are t w o

variations

of this

method:

One

updates

the

inverse

of

the

. J acobian matrix; the other approach updates the . Jacobian matrix. A . Jacobian

matrix contains

all

the derivatives of the equations with respect

to

unknowns.

Since a

set of

linear

equations

is

solved

repeatedly

in

the

proces s of s o lv ing

a

set of

nonlinear

equations,

updating

the

inverse

of

the

Jacobian matrix

results into

an

easier task

i n s o lv ing

the linear equations. Only a straight

forward

matrix multiplication is

required.

Otherwise,

either an iterative

approach or

a

direct method is

needed. The

variation

wh ich

updates

the

Jacobian

matrix

performs

better

in

practice.

This

is

due

to the instability

of

the update

equation

for

the

inverse of the

Jacobian matrix.

In

trying

to improve the reliability of this method,

Pa los ch i

and Perkins [39]

developed

several

update procedures for the Jacobian matrix. Their

modifications

have b een


23/148

10

used successfu l ly

to

solve many chemical engineering

and mathematical

problems.

One other attractive

numerical

method

is

the

we l l - known Powell's

d og le g method.

This approach takes a hybrid of Newton's method and the steepest descent method.

The motivation for this concept is to

take

advantage of the global convergence b e

havior

of

the steepest descent method and the loca l

quadratic

convergence behavior

of

Newton's

method. In Hiebert's

evaluation

of mathematical software fo r so lv ing

systems of

nonlinear

equations,

he

concluded

that this

approach did not perform

better

than

the

quasi-Newton method

[2.3].

Chen

and

Stadtherr [8] mod i f i e d P owe l l ' s

method b y adding an

automatic

scaling step for both functions and variables, a n e w

test

for

nearby loca l minima, and a p r ov i s ion to force unknowns to be

nonnegative.

The last addition

is

logical since most of the var iables involved

in

chemical engineer

ing problems are

nonnegative.

With these modifications, they reported

a

remarkably

improved performance

of

Powell's

d og le g method.

In

this

approach, Broyden's u p

date

for

the

Jacobian matrix

is

used. The n e w update introduced b y Paloschi and

Perkins

[39]

can

also

be

used.

In sequential modular simulators, the

nonlinear

equations are dense, implying

that the elements of the Jacobian

matrix

are

mostly

nonzero. On the contrary, the

nonlinear equations that arise

from

equation-based simulators produce

a

very sparse

. J acobian

matrix.

A s reported b y Stadtherr and W o o d [58] , the percent o f nonzero

elements in

the Jacobian matrix

is mostly below

ten.

Maintaining the sparsity

of

this

matrix

is

important

since only nonzero

elements

are stored. In addition, if

the

sparsity

degenerates,

w e

may run

out of computer

memory

in the p r oc e s s of

solving

the set

of nonlinear equations since

the cardinality

of the set

is

always ve r y

large.

In order to

maintain the

sparsity of

the

Jacobian

matrix,

a ne w

update formula


24/148

11

was developed

b y Schubert

[53]. With inclus ion of this update formula, Chen and

Stadtherr [9] successfu l ly

so lved

many f l owshee t ing problems. In contrast,

other

researchers reported a very poor performance of Schubert's update. More recent ly ,

Bogle and

Perkins [ - 5 ] der ived

a

n e w update formula for

use with

the

sparse

Jacobian

matrix. Sun

and

Stadtherr [64] incorporated this n e w update formula and reported

promising results for their vers ion of Pow el l 's dog leg

method.

Since most of the successes of the modi f i ed Powel l ' s dog leg method were re

ported by the original authors, w e investigated this numerical method extensively

for

further

clar i f ica t ion of its

performance

in so lv ing

both sparse

and

dense sets

of

nonlinear equations. W e a l s o tested two

approaches for

keep ing the u n k n o w n s within

prespec i f i ed bounds.

Tearing Algorithms

Flowshee t

partitioning

and

tearing

is

an

integral

part

of

sequential

modular

sim

ulation. In the process

of

f lowshee t partitioning and tearing, a computation sequence

and

the associated set of tear

streams are

generated. The subject of

partitioning and

tear stream se lect ion has emphasized the identification of the optimal

tear

set for a

given flowsheet. The objective in identifying

an

optimal

tear set

is

to minimize

the

required

computation

time in the

actual simulation. There

have been seve ra l criteria

u sed

to

characterize

an optimal

tear

set.

The main

criteria are

1. Minimize

the number

of iterated variables

2 . Minimize the sum of

stream

weigh t s in the tear set

3. Minimize

the number

of

times

loops are torn


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12

A tear stream se lec t ion

algorithm

that

sat isf ies the above three

criteria

may not

minimize the computation time espec ia l ly when des ign constraints are p r e se n t i n

the

flowsheet.

This is because the

criteria listed

above are

sometimes mutually exc lus ive

and constraints have va r y ing effects on

the

c onve r ge nc e behavior. Gros et al.

[21]

showe d

that

the numerical method used to converge the

tear

variables d oe s not

affect

the c onve r ge nc e behavior of an unconstrained flowsheet. It wil l be s h o w n later, in

Chapter

4 of this

thesis,

that the

c onve r ge nc e behavior

of

a constrained f l owshe e t

i s

sensitive

to the c onve r ge nc e method

use d

in

converg ing

the tear streams and it is

a lso

a f fec ted

b y

the

type

of

constraints.

H e nc e ,

formulating

a

reliable

tearing algorithm

is not

as easy

as it

first appears. Since most commercial simulators use sequential

modular computation, p r ov id ing an efficient partitioning

and

tearing algorithm

is

essential in helping c he mic a l engineers

simulate

practical flowsheets.

A t

the

f l owshe e t partitioning

step,

each strong component in a f lowshee t

is

iden

t i f ied.

Within

a

strong component, there must be a zero

or

nonzero length path

from

any one node (A)

to

al l

other

nodes including

node

A

itself.

Unless

the

strong

component consists

of

only one s ing le node,

w e

a lway s have

a cycl i ca l ly

connected

graph in a strong component

s ince

w e

can a lway s

revis i t the node

where

w e

began

b y traversing the graph thru a wel l -def ined path.

In order

to

make

the graph acyc l ic ,

some streams must be

se l e c t e d a s

tear streams.

This

is

the

tearing

step or

simply

the tear

stream

se lec t ion

step.

M o s t tear stream se lec t ion algorithms

w o r k

with

the

cyc le

matrix

where

al l l oop s and streams participating

in

the loops are

stored

in a

matrix.

For

f lowshee ts

with

just a few cycles , identifying all loops

is

trivial. A s noted

b y Gundersen

and

Hertzberg

[22],

this

step

can

be

very

expensive

when a ve r y huge

number of cyc les exist in

the

flowsheet. In addition, the required storage for such

a


26/148

13

cyc le

matrix may be

v ery

large. For the heavy water plant presented b y Gundersen

and

Hertzberg

[22] , the storage requirement for the

fu l l

cycle matrix

amounts

to ap

proximately

4.5

MB assuming a 2-byte

integer

space for

each

element of the matrix.

We we r e

not able

to

find

a tear

set for this

problem using

ASPEN PLUS

[1]

on

a

DEC 3100 due to the lack

of memory.

W e

conclude that

tearing algorithms ut i l iz ing

the

cyc le matrix

are impractical w h e n a large number of cycles and units are present.

John and Mii l le r [22] deve l oped an algorithm that can produce an

optimal tear

set as the tear set

that

minimiz es the sum

of

tear

stream weights . In

their

algorithm,

a

branch and bound method was u s ed to

reduce the dimension

of the

combinato

r ial problem. This approach

requ ires

bounds on tear sets of

v ar iou s

subproblems.

An

eff icient algorithm must be

av a i l ab le

to

produce such bounds for the success of

the branch

and

bound method. An

exce l l en t r ev iew on this

subject

was written b y

Gundersen

and Hertzberg [22]. A s

discussed above, w e

can not

guarantee

that this

minimum tear set

requires

the minimum computation time. Furthermore, if the flow

sheet has many

cyc les l ike

the

heavy water plant,

a

simpler approach

that

produces

a close-to-minimal

tear

set without requiring large storage

space is

acceptable.

Gundersen

and

Hertzberg

[22]

introduced

a tearing

algorithm that

does not

use

the cycle

matrix. Since

it is based on a simple heuristic rule

that

se lec ts the

input

streams

to

a unit that produce the most output information as tear

streams,

it

does

not consistently produce an optimum tear set. They described it as a c los e - to -

optimal"

tearing algorithm. More

recent ly ,

Li

et al.

[31]

developed

a

n e w

algorithm

that

also

does not

use the cycle

matrix. We implemented their algorithm

and fou nd

that

it also fa i led to

consistently

giv e an optimum tear set. Its

re l iab i l i ty

in finding

an optimum tear set was found to be

similar to that

of Gundersen's algorithm. Lien


27/148

14

and Hertzberg [30] modi f ied

the

original Gundersen algorithm

b y

introducing a new

tearing

criterion

that

uses the

cyc le

matrix.

Their

reported results s h o w e d that the

n e w

algorithm produced the optimal

tear

set

for cases where

the

original algorithm

had fa i led . As d i scu ssed by

Lien

and

Hertzberg

[.30], this

is caused

b y the l ack of

a tie r e so lu t i on scheme in the

algorithm

itself. The improvement w as

obtained

by

sacr i f ic ing data storage

space

and

speed

since

the cyc le

matrix has to be updated

repeatedly. Since

w e

are interested in tearing algorithms that do not u se the cycle

matrix, the i m p r o v e d Gundersen algorithm was not cons ide red here .

In

this work ,

w e

propose

a new

tearing algorithm

based on

the

concepts of Li 's

algorithm.

Densities

from Equations of State

In

process

simulation,

obtaining the liquid and

v a p o r

densities

of

a mixture

constitutes

the

fundamental

step

in obtaining

other

properties required

for

phase

equilibrium and material and energy

balance

computations. O f

course,

there are

in

stances

where

other

correlations are u sed

to compute

des i r ed properties. Generally,

the speci f ica t ions of the state of a

mixture

with

a

spec i f i ed composition

are

temper

ature and pressu re

for the

simple reason that

these

two intensive var iab les can

be

measured read i ly . For a pressure explicit equation of state, so lv ing for

the

densities

of

vapor

and

l iqu id

is

then an iterative

process.

Most

equations

of

state are

a

third

order

polynomial

in

density.

This implies that the equations of

state

are not

mono-

tonic throughout the range

of

interest.

Depending on the specif ied

state, w e

may

have one density root or

three density roots

where the maximum and minimum roots

correspond

to

l iqu id and

vapor

densities, r e spec t i ve ly . The

other

density root

is

in

a


28/148

15

phys ica l ly unapproachable r e g ion . Mos t iterative numerical methods have been d e

ve lop e d

by

assuming

the

existence

of

monotonie be hav io r

of

functions. This certainly

imposes a constraint to

the

applicability

of

these methods in so lv ing

for the

densities

of an

equation

of

state. Another

w a y of so lv ing for the densities is to

use algorithms

spec ia l ly tailored for

obtaining

al l

roots

of a polynomial. Ho w e v e r , w e don't need

both l iquid and vapor densities at

the

same time. Consequently, iterative methods

are still

the

prefe r red route for

determining

the

density from

an

equation of state.

Mathias et al. [ . 3 5 ] developed

an

alogorithm for obtaining the desired density

from

an equation

of

state.

This particular algorithm

was

use d

in the

phys ica l

proper

ties

system

of

ASPEN. More recent ly ,

Topl i ss

[65] published a new algorithm that he

contended

was more eff icient . Sinc e pressure,

temperature

and composition

of

a m i x

ture

are common iterative var iables in

the

solution

of

f lowshee t ing problems,

w e

may

encounter cases where no rea l density

root exists for

both

vapor and liquid. Topliss's

algorithm

stops

at

this point if such a situation is determined. In contrast, Mathias

et al.

[35]

produced

a

pseudo-root

for

this

c ase

and let the simulation

calculations

continue.

This is

desirable because such a phenomenon normally

occurs

in the midst

of

so lv ing

the

f lowshee t ing

problem; the temperature,

pressure

and composition of

a mixture

can

be ve r y

unreasonable for

properties calculations.

If

a pseudo-root

is

used, w e may get to a val id spec i f ica t ion in

the

next iteration of

the

computations.

Care must be exerc ised to check

the

val id i ty of

the

f inal

simulation

solutions.

W e

implemented

Topl i ss ' s

algorithm

and

extended

it so

that

pseudo-roots are

generated

when inva l id spec i f ica t ions

occur.

Our present phys ica l

property system

uses

this extended algorithm

as the underlying

method

for

determining densities

from

an

equation

of state.


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16

Object Oriented Programming

Modern software engineering fundamentals stress

the

notion

of

data

m o d e l i n g

in

the

development

of ne w

software

instead of functional

analysis

[36] .

Modular

ity

of

software is a lways

emphasized. Unfortunately, no

one

clear

def ini t ion

for

the

meaning

of

modularity is suff ic ient .

M e y e r

[ . 3 6 ] suggested

that

a comprehensive

def

inition of

modularity

should address

var ious aspects of

good software qualities l ike

extendibility,

reusab i l i ty ,

etc. In

l igh t

of dev e lop ing modular

software,

object ori

ented programming has arisen as one

of

the

prominent routes towards modularity.

S ince most

proces s

simulators have complex requirements

and intricate

relationships

among the constituent elements,

w e

need

a systematic approach

towards

the des ign

of proces s

simulators. Object oriented programming concepts facilitate

such

a sys

tematic

approach

in the proces s of

software

construction. M o s t importantly, they

emphasize

data

modeling and

system des ign whic h promote software

extendibility

and

reusab i l i ty .

These qualities are crucial

in

a

proces s

simulator.

D e v e l o p m e n t of

a

versatile

proces s

simulator is inherently an

o n - g o i n g

task. N e w

capabilities must

be incorporated into the simulator as

they

arise

without

causing

any drastic

change

in

the

simulation

system.

Also , modifications

of

old algorithms implemented previ

ous ly

should not affect

the

continuing user.

Current

modules should

be ut i l ized in

future development.

All

these requirements

point towards

the use

of

object oriented

programming notions in the development

of

a

proces s

simulation environment.

An object consists

of

data and operations that

manipulate

or act

o n

the data.

S of tware construction thru object-oriented

programming is

es s en t i a lly iden t i fy ing var

ious objects

required in a

spec i f ic application

and

def ining

relationships

among the

objects.

Hence, the notion

of

data modeling

is

just the

proces s of i den t i fy ing

objects


30/148

17

and

def in ing their

respective

relationships.

The operations def ined

must

be

ade

quate to

support

sof tware

requirements. Encapsulation is

the

creation of a

boundary

around

an

object.

In other words, only

operations def ined

for an object can

act on

the

object. This

gives

us the ability

to

limit

the

use of objects

so

that they will not

be m i sused

unintentionally

like the COMMON

block in

Fortran.

Information hiding

is a direct

benef i t

that comes from encapsulation.

Si nc e an

object's data is available

to

the

outside wor ld

only thru its operations,

programmers c an

hide implementation

details from the user. For example, when w e h a v e a set of numbers in an

object,

the

operation

to

produce

an ordered list

can

be implemented in severa l ways .

The

user

k nows

only that the

operation returns an ordered l i s t and nothing else. In

fact,

this

notion is not new , w e

can

also obtain

information

hiding from conventional program

ming languages l ike Pascal, Fortran, C, etc. How eve r ,

encapsulation

does

more

than

just information hiding b y imposing constraints on operations permitted

to

ac c e s s

the

data

of an object.

Mes s age sending

is the

w a y objects communicate

with the

outside wor ld

thru

def ined

operations. We

send

a message

to

an object and it

will

then decide what

to

do with

that m essage . This is parallel

with

the

conventional

procedure ca l l s .

The

procedure is the operation associated with an object; the procedure's argument

l i s t

is the data

of

an

object.

Clearly,

w e

have a reverse ownership

of

argument

l i s t

and

procedures

in

object oriented programming.

Her e ,

the data ow n the

operations.

Importance

of

data

in object

oriented

programming

is

an

aspect

which should

not be

over looked .

Inheritance is the

ability

to

create new objects (derived objects) b y inheriting

data and

operations of

previously def ined

objects

(base objects).

Conceptually, w e


31/148

18

can v iew the b a s e object as a gene r a l i zed object and the

der ived object

as a s pec ia l i z ed

object.

For

example, a complex number has a rea l part and a complex part.

The

rea l

part i s

a

gene r a l i zed object since

every

rea l

number

has

this

part.

The

complex

part

is

a spec ia l i zed object because only a complex

number

has this

part. The

rea l v a lu e

of

inheritance is in software development and espec ia l ly

in

data abstraction. To the

user of an

object

oriented application so f tw a r e ,

inheritance is

almost invi s ib le .

From a

programming perspective,

inheritance p r ov i de s

for the reduction of code duplication

s i nc e a spec ia l i zed object inherits

attributes

from fu l ly deve l oped

and tested

objects

and

their

operations.

The

program

structure and information

m o d e l i n g

of

this work are

described

b y

Varma [67] . Examples manifesting the benef i ts of object oriented programming wil l

be shown in the d i sc us s i ons of numerical procedures.


32/148

19

CHAPTER

3. NUMERICAL

PROCEDURES

Numerical

Methods

The

problem of f inding a

so lu t i on

to

a

set of

nonlinear equations

can

be

def ined

as: g iven

f(x)

= 0,

w e

would l ike to find a vector x* such

that

f(x*) ^ 0 or

f(x*)

~ tolerance. M o s t methods ava i lable

for solving

this problem are iterative.

They

generate

a sequence of

vec tors

{x^} and

if

the method

succeeds, then

{x^}

wil l co n v e r g e to {x*}.

Newton s method

Fo r

the

class ic Newton's method, the iterative

wor k ing

equations

for

so lv ing such

a set of nonlinear equations a re:

= H-+Pk (3 II

H^k'lPk

=

-f(x&l ( 3 - 2 )

To

so lve fo r pj^, w e h a v e to

solve a se t

of

linear equations

def ined in equation 3.2.

Standard

Gaussian

elimination with

p ivo t i ng

is

usually

used

to

ac c omp l i sh

this

task.

Hence, for

an initial {xo}, w e ne e d

to

compute

the

Jacobian

matrix.

J(xo), and then

calculate pj^. Subsequently,

a

n e w {x} i s generated. This p r oc e s s

continues

unt i l l

{x^} co n v e r g e s to the

solution within

a speci f i ed

tolerance.

Note that the Jacobian


33/148


34/148

21

There h a v e

been

t w o

approaches to this initialization problem.

One

is to

use finite

difference

as

for

the

cas e of

Newton's

method.

Since

the initial guesses , {xo}, can

be ve r y far

a w a y

from the solution,

it may

not

be neces s a ry

to

p r o v i d e

an accurate

approximation

of

the

Jacobian

matrix. With this

argument, an

identity matrix is

u se d as the initial . J acob ian matrix. In

practice,

both approaches

h a v e

b een u s ed

successful ly.

But, u s ing an identity matrix to in i t i a l i ze

the

Jacobian matrix

can

cau s e

the

method

to perform

poorly in

some

cases .

Modified

Powell's dogleg

method

This method is

a

hybrid of

Newton's

method and the

steepest

des cen t

method.

The

der iv a t ion of

this

method

can be found in a paper

b y

Chen

and Stadtherr [8].

The

work ing equations can

be summarized as fo l lows:

(3 . 1 0 )

(3 . 1 1 )

( 3 . 8 )

( 3 . 9 )

p" =

Dip

(.3.12)

Dy

and

are function

s ca l ing and

variable

s ca l ing

matrices,

g is the s ca led steepest

des cen t

direction.

Afte r

computing

p ^ ^ and

p ,

then

Powel l ' s

search d i rec t ion

is

determined according to

the

fo l lowing equations:

P = P^^

if

^

> I I P ^ ^ I I

p

=

ap^

+

(l-a)p^

if llp^^ll

> A > l | p - | |

(3 . 1 3 )

( 3 . 1 4 )


35/148

22

P = yr^g if I 1 P ' 1 1 >^ ( 3 .15 )

llgll

w h e r e a is def ined in the fo l lowing equation.

" (piv_pi')V

+

r

r =

{1(p ^) P -A2|2

+ [||P V||2_a2][A2-||P||2]}1/2

, 3 . 1 7 ,

A is the radius of the reg ion

within whic h

the

linearization can

be trusted.

For initial

guesses

X q

and

f(xo), an algorithm

of this method

can be def ined

as

fo l lows:

1. Calculate the Jacobian matrix

by

f in i t e

di f fe rence .

2.

Calculate Dy

and Dx and

s ca le the

Jacobian matrix.

Dy

is c hose n

s u ch

that

the largest

absolute va lue

in

each

row of the

matrix DyB

is equal

to uni ty .

S imi la r ly , ch oos e Dj; s u ch

that

the l a rges t absolute v a lu e in each co lu mn of

Ds

qual to

u ni ty .

3. Calculate initial step bound, A = r * maa;{ | | Da ;X o | |, 10 .0 } . r is a number

provided

by the

user. A s the def in ing equation

shows,

it affects the va lue

of

the

initial step

bound.

4.

Calculate the s ea rch

direction p},

accord ing

to above d e f ine d equations.

5. Evaluate

+

pj^.)

6.

Check

for s low c onve r ge nc e or nonconv ergence . In this step,

a

n e w

test

for loca l

minima w a s p r o p o s e d b y Chen and Stadtherr [8] .

7. Check

if Jacobian matrix needs

to be re-evaluated.


36/148


37/148

24

In

testing our implementation,

w e u sed ten chemica l eng inee r ing problems

p r o

p o s e d b y Shacham

[56].

These

problems

were a l so used by Sun and Stadtherr

[64].

The problem description is shown in Table

3.1

and

the

details of the problems are

included in

Appendix

B.

Table 3.1: Test

problems

for

fu l l

matrix

modi f i ed

Powel l ' s

method

Problem

D i m e n s i o n Description

1

2

Material balance of

a

reactor

2 7

Chemical equilibrium

of

o x y g e n

and methane

3

2

Thermodynamics

of

a

2-component l i qu id mixture

4 2

Equilibrium

c onve r s i on

of a reactor

5 2

Material and e n e r g y balances of

a

reactor

6

2

Chemical equilibrium problem

7 13

F r e e

energy minimization

of a reacting

system

8

6

Steady

state

kine t i c s

9 6

Chemical

equilibrium

problem

10

10

Combustion of

propane

and air

In

these problems,

there are

var i ab l es

that are

nonnegative.

For

example,

the

composition

and flowrates h a v e

to

be

positive. H o w e v e r ,

there are var i ab l es that can

be

positive and

negative

in chemical eng inee r ing problems. Enthalpy and heat duty

are

two

o b v i o u s examples. W e implemented

two

w ays of ensuring var i ab l es to be

positive. The first approach i s

to

take the absolute

va lu e

of

the negative va lu e

and

adjust

pf,

accord ing ly . The

result

of this

step is abandoning the

direction of

current

iteration and

starting

at a n e w g u e s s

point. One

other

w ay i s

to

keep

the

direction

of

by imposing a step bound

factor

just l ike the

case

of damped Newton's method.

Instead of minimiz ing the norm of

the

functions, this factor is u sed to keep va r i ab l es


38/148

25

in the def ined

bounds. The

equation for

computing

the factor i s

given

b e low.

A

=

Vimin

*

(3 .22)

P f

where a i s a small number

close

to one

and

xp

is

the lower bound of var i ab l e Xj .

Note that the computation of A i s only done for

all

p^''s that are l ess than ze ro .

Q is used

to

ensure that the constrained variable will not

reach

exactly

the

lower

bound but very c lose to

it. This is

because w e

may have

terms in the equations that

are

undefined

if some var iab les are ze ro exactly. F o r example, the logarithm of a

composition var i ab l e with a value of z e r o

w oul d

cause an error.

We

used 0 .99 for a

in our code.

In Sun and Stadtherr's

evaluation of numerical

methods, they

c o n c l u d e d

that

the modi f i ed Powel l ' s

method is more reliable

than

the quasi-Newton method

that

uses

Broyden's

update [64]. The

results of our investigation are s h o w n in Table 3.2.

All of the

problem

numbers with

alphabetical

endings represent runs with different

initial gu esses

for

the same problem. A n /in the

table

indicates a fa i lu re in obtaining

the f inal solution. Some of the

problems

are numerically singular at the given initial

guesses.

These problems w er e so l ved with two scaling options, function scal ing (FS)

and both function and var i ab l e sca l i ngs (FVS).

The t w o b o u n d - c h e c k i n g

strategies

w e r e a l s o tested. There is

a

use r g i ven

parameter,

r, in the algorithm but Sun and

Stadtherr

did

not

report

any values of

r

u s e d in their

runs. For

our results, al l

the

problems w e r e

so l ved

with the fo l lowing

set

of va lu es and the best

performance is

reported.

r

= {0.1,0.2,0.3,...,1,5,10}

The results

obtained

c lear ly

support

the re l iab i l i ty of this

method

as reported

b y


39/148

26

Sun and Stadtherr [64]. Both of

the

approaches

for

keep ing u n k n o w n s in

the

feas ib le

r eg ion work ve ry well .

They do not

impede the progress of

the

numerical

method.

The

method

that abandons

pjr.

with

only function sca l ing fa i ls

5

times.

Only 4

cases

fa i led

fo r

the

other

va r i a t i ons in this test. This gives a success

rate

of more than 80 %

for

al l

of the

different

va r i a t i ons of the modi f ied Powel l ' s dog leg method

tested.

The

results

a l so indicate that the performance of the

method is not ve r y much af fec ted

b y

variable sca l ing . For th i s set

of

test problems, there i s no one clea r optimum va lue

of

r. Ho w e v e r ,

one

was found

to

be a good number

for

mos t

of the

p r o b l e m s studied.

The

algorithm

for

so lv ing

a

set

of

sparse

non l ine a r

equations

is

essent ia l ly

the

same as

the ful l

matrix c ase except for

the fo l lowing

steps:

1.

Evaluating

the Jacobian matrix.

2. Solv ing

the

l inear se t of equations.

3. Updating the Jacobian matrix.

In evaluating the Jacobian matrix for

the

ful l non l ine a r equations, every s ing le

unknown

is

perturbed i nd iv id ua l l y . Obv ious ly , there are

many unknowns

in most

sparse systems of equations.

U s i n g the ful l

matrix approach

is

too

cos t ly .

Further

more, there are equations that may

not

be af fec ted b y the perturbed variable. Sun

and Stadtherr [63]

studied

two

algorithms

for simultaneous perturbation

of

unknowns

and co n c l u de d that both are effect ive . These t w o algorithms w e r e d e ve lop e d b y Cur

tis

et

al.

[12]

and

C o l e m a n

and

More

[11].

The

main

thrust

of

these

two

algorithms

is

to

group

var iab le s

which invo lve

di f fe ren t

functions

as a cluster. Then these variables

can be perturbed

simultaneously

when computing

the

Jacobian matrix. Considering

the

oc c u r e nc e matrix in Figure 3.1, w e can group

{21,22} ^^d

{23,2^,25} as two


40/148

27

Table 3.2 : Number

of

iterations for ful l matrix modi f ied

P owe l l ' s method in test

runs

Preserve

P A .

Abandon

Pk

Problem

FVS

FS

FVS

FS

1

13 17

1 2

15

2 a

47

46

3 2

38

2 b 2 0

20

18

22

3

10 10 1 0

1 0

4a f

f

f

f

4b 12 15

1 2

15

4c

1 0 11 10 11

4d 21

25

2 1

25

5a

7 8 8

7

5b 5 5

5

5

5 6

6 6 6

5 d 19

1 4

19

1 4

6a 16 1 6 1 6 1 6

6b 2 5 3 2 2 5

32

7

40

5 6 3 3

28

8a

7 7 7 7

8b

f f

f

f

8c

8

8

8 8

8 d f

f

f

f

8 e 4 9

17

36

18

120 88

1 6 3 f

9a

f

f

f

f

9b 3 6 36

40

42

9 c

4 0 4 0

2 6

26

9 d 4 9

49

3 7

37

1 0 a 2 4 2 4

22

22

1 0 b

2 0 2 0

15

15

10c

38

36

3 5

36

lOd 44 39 41

40


41/148


42/148

29

(b)

Perturb each var i ab l e in the variable set separately and evaluate the func

tion.

2. Repeat step

1

until al l

equations are p r o c e s s e d .

It

i s

clear

that

this strategy computes the Jacobian matrix one

r ow

at a time.

This

approach is

desirable only if al l the

equations

are simple equations. For

instance,

they don't need

to

cal l some other subroutine that accepts parameters that may

or may not i nvo lve the

unknowns

in order to compute the function va lu e . The

majority of the equations in process simulation fal l

into

this

category.

There is

another

category

of equations

that need

to

make subroutine

cal ls before thay

can

compute

the respective

function values . In phase

equilibrium

calculation, the A'j

va l ue s

are good examples

of

this type

of equations.

Before

computing

A'j , the fu gac i t y

coeff ic ient

of component

i

fo r the l iqu id

phase

and v a p o r

phase must

be computed.

These calculations

n e e d

l iqu id and v a p o r

c o m p o s i t i o n s

that are normally

u n k n o w n s

in the set

of

equations. Mathematically, the equations are as

fol lows :

K i - g ( T , P , X , Y )

= 0

: :

( 3 . :% )

K n

- g

T , P

, X , Y )

= 0

With just a

function

cal l to the procedure g ,

w e

obtain al l

values .

Hence ,

w e

can

group

al l

these

equations

into

a

procedural equation

which

ca l l s

the

procedure

g

and produces a sequence of

function

values . This procedural equation a l so

has an

associated set of

var i ab l es for Jacobian matrix

calculation.

F o r

this example, this

se t is {A'j,..., A'n,

T,

P,

X,

Y}. In

this manner, the Jacobian

matrix is evaluated

column wise instead of row w i se as in the case of simple equations. With

the

addition


43/148

30

of procedural equations, the Jacobian matrix is f i l led up column wise

and

row wise

interspersely depending

on

the equation type.

Fo r

the

solution of a set of sparse linear equations,

the

main

concern i s to pre

se r ve the

sparsity

of the matrix. Ideal ly,

if a

matrix i s triangular, no

f i l l - in

i s poss ib le .

A fi l l - in i s a

generated nonze r o e l e me n t in

the

p r oc e s s

of

so lv ing

the equations. C o n

sequently,

f i l l - in ' s

occur only

thru

deviation

from

a triangular

matrix.

Sinc e a set

of

sparse

linear

equations has

to

be

so lved repeatedly, efficient algorithms which exp lo i t

the

sparsity of

the matrix must

be used .

Standard Gaussian

elimination

causes the

sparsity

to

degenerate.

Stadtherr and Wood [.58, 59] developed severa l

algorithms

that

can

so lve a set of sparse l inear equations efficiently . Their algorithms cons i s t of

t w o major

steps:

1.

Reordering phase

2 . Numerical phase

An algorithm

that

solves

a

se t

of

linear equations in

t w o

phases

i s ca l l ed

a

tw o -

p ass approach.

There

i s another method c a l l e d a priori approach

which s w i t ch e s

between reordering

and

nume r i c a l steps as

the

linear

equations

are be ing so lved.

Stadtherr and W o o d

[59] co n c l u de d that

the t w o - p a s s approach i s

superior

to the a

priori method. In

the

reorder ing phase,

the matrix

is permuted to

block

triangular

form. A s

its

name impUes , the nume r i c a l phase

is

the

task

of so lv ing

the

equations. A t

this stage,

if

the

d iagona l e l e me n t

i s

not

a

suitable

pivot, column or

row

interchange

wil l take place. This o b v i o u s l y

defea ts

the

purpose

of the reordering phase

be c ause

the matrix is ordered

such

that

min ima l f i l l - in ' s

are

generated in

the

nume r i c a l

step.

H e nc e , maintaining

the sparsity of

a

matrix and establishing the stability of the


44/148

31

s o lu t ion

p r oc e s s

are

mutually exclus ive . To reso lve this dilemma, a threshold

p iv o t ing

strategy

is used. Here,

p iv o t s

are

considered

acceptable

if

they

are l a rge r than

some

k X max, where&isa constant less

than

1

and

max is the

largest

e lement

in a column

or

r ow.

The v a lu e for

k

i s n o r m a l l y v e r y

small.

A

commonly use d v a lue is 0 .1 . If k

approaches 1,

then

this method reduces

to

the normal

p iv o t ing

strategy.

W e implemented Stadtherr and Wood's algorithm for so lv ing a

set

of sparse

linear

equations. The

details

of the

algorithm

are described by M ah [33]. S ince three

algorithms we r e

p r e s e n t e d

b y Stadtherr

and

Wood

[58 , 59] , we opted

to

use SPKl

for

the

reordering

phase

and

RANKI for the

numerical

phase. This combination is

among

the

better

per fo rmers in

their tests. Before applying these two algorithms, the

matrix should

be

partitioned

o r

reordered

into

a block-triangular form.

Partitioning

cons is ts of

two

steps. Ini t ia l ly ,

the matrix must

satisfy

the

cond i t ion

of maximal

transversal. That is ,

the matrix

must

have

a

set

of

z e ro

f ree

diagonal elements.

An

efficient algorithm to obtain this property

for a

giv en matrix was

presented

by

Duff [15] .

This algorithm

was u s ed

in

our implementation.

The

next step,

a

simple

symmetric

permutation of

the matrix

wil l

produce a block-triangular

f o r m

matrix.

The permutation

can

be

done

eff icient ly

with

the

we l l - known

Tarjan's

algorithm.

Duff and

Reid [14]

prov ided

a ve r y good implementation

of

this algorithm.

W e

coded

this algorithm

b as ed

on the wor k of Duff and Reid

[14] .

Note

that this algorithm

a l s o

s e rv es as a strong component

finder

in our studies

of

tearing algorithms

to be

pres en ted

in

the

next section.

Fina l ly ,

for

the update of the sparse

Jacobian

matrix, w e

studied

two update

equations presented

b y

Schubert

[53] , and

Bog le

and

Perkins

[5] .

Schubert's

update


45/148

32

is

def ined

as:

B&+1

=

Bi.

+ Z{D(yj-Bi.pfc)p[.}

( 3 . 24 )

Z

is an operator

def ined fo r

any matrix

M

by:

Z{M}

=

0

i f Y i j =

0

= M i j i f Y i j =

1

( 3 .2 5 )

w her e

Y

i s the occurence

matrix of

the set of sparse nonlinear equations to be so l ved ,

Yij

is

the

element in

row

i

and

column

j

of

Y.

If

w e

let be

the

vec to r

formed

from by

setting the

component to

zero

if

Yij

i s

zero, then

D

=

(3.26)

I 1 ^ 7 7

Each element in

D i s de f ined

to

be

1/a^

if a *

is

not zero ;

otherwise

the element is

as s i gned to be zero, a is

the

dot

product of the

v ec to r p^ w i th i tself .

The

update

i s

des i gned to operate

b y rows

in order to p r e se r ve

sparsity

of the

Jacobian

matrix.

When

the

diagonal e l em en t

Djj

is too small or zero , then row i

is

lef t unchanged.

For

Bogle ' s update,

the equation

is as

fo l lows:

yf

= )-%(%&)

This equation a l s o pe r f o r m s update b y rows . Si m i l a r action

is

taken if the d e n o m i

nator is too small. N o

update

is done to

that

particular r ow . Note

that

if any B^-

e l em en t

is

zero , the sec ond term

will

a l so become ze r o due to (B^ )^ . Therefore, the

sparsity

is

p r e se r ved .


46/148

33

In

testing

our sparse nonlinear equation

so lver ,

ten problems w e r e so l ved .

The

description of

these

problems is

g i ven

in Table

3.3 . The

results obtained

are

presented

in Table 3.4.

The

problem numbers with alphabetical endings represent

either

s imi la r

problems

with

dif ferent initial guesses o r with dif ferent input

parameters

for

the

equations.

For

instance, the

split f r ac t i on of

a simple splitter

is

changed. The

bound

checking strategies

used for the fu l l matrix

are a l so included

in this

test

s i nc e m os t

of the

var i ab l es are

nonnegative. Similar ly ,

the

symbol

FVS stands

for

both

function

and variable sca l i ng , and FS indicates only function sca l i ng . For the

v a lu es

of r,

w e used {0.1,0.2, ...,1,5,10}. The r e su l t show n

is

the best from these 12 dif ferent

v a lu es

of r.

Table 3.3; Test problems for sparse matrix modi f ied

Powel l ' s

method

Problem

D i m ens i on

Descr ip t i on

1

12

Isothermal f l ash

us ing composition

2

12

Isothermal f l ash

us ing flowrates

3

30

Si ng l e l oop f l ow shee t

material

balance

us ing

compositions

4

30 Si ng l e l oop

f lowshee t material

balance us i ng flowrates

5 10 0

Schubert's test problem with

dif l 'erent

dimensions [53]

6

17 0

Cavett p r o b l e m

7 19

D e w point ca lcula t ion

8 19

Bubble point c a l c u l a t i on

9

19

Adiabatic

f l a sh c a l c u l a t i on

10

4 3

Ammonia f lowshee t material balance

The

results

s h o w ve r y

good

pe r f o r m anc e

of

modi f ied

Powel l ' s

dog l eg

method

with both Schubert's update and

Bogle ' s

update.

In

the case of Schubert's update,

a f ew r e sea r c he r s reported

ve r y

p o o r performance of this

formula

when

they

u s ed

it in place

of

Broyden's formula in solv ing a sparse set

of

nonlinear equations. N o t

surpr i s ingly ,

Chen and Stadtherr

reported p r om i s i ng performance

of

Schubert's up -


47/148


48/148

35

date. This

may indicate that

the p o o r performance is caused

b y

the w ay

the sea rch

direction (p^) is

computed.

Bogle ' s update a l so g i ve s v e r y encouraging results. This

is

also

in

agreement with the r e su l t s of Sun

and

Stadtherr. Again,

the

results s h o w

not much degradation in

p e r f o r m a n c e if

variable

scaling i s

omitted. Neither bound

check ing strategy impedes the progress

of

the so lve r .

The best

r va lu e

var ies

drasti

ca l l y for this set of

problems.

Some problems w e r e c o n v e r g e d requiring the minimum

iterations

with a r

va lu e

less than one.

There

w er e

a

f ew

cases

ach i ev ing

convergence

with

minimum

number of

iterations with a r va lu e

of

larger than one. N o

apparent

optimum

va lu e for fis found. Although

the

va lu e of r is not reported here, it should

be noted that the number of iterations

i s

quite sensitive to the va lu e of r. This

is

p o s s i b l e because r determines the initial trust

region. This

may l ead to taking

the

wrong direction

in i t ia l ly

b y

not interpolating

between the

directions of Newton's

method and

the

steepest-descent

method. Then the trust r eg ion

can

not recover to

guide the direction of interpolation step.

We

observed that for seve ra l fai led cases ,

the trust

reg ion

w as very

small implying that

only the

steepest descent

d i r ec t i on

w as

used in the solution process . The algorithm detected that c o n v e r g e n c e

was

too

s low

and the iterative

calculation

stopped.

We

attempted to

overcome

this p r o b l e m b y

switching

to

Newton's method after

seve ra l

consecutive steepest-descent

iterations.

For

some problems, this

appeared

to

be he lp fu l . Unfortunately,

this

quick

fix a l so

fai led

for some cases .

C onsequ en t ly , there

is

still no

easy w a y

of avo id ing taking

too

many

steepest-descent steps.

In

summary,

w e

f o u n d

that the modi f i ed Powel l ' s dog leg method

i s r e l i able

in

so lv ing the ful l and sparse systems of

nonlinear

equations. The fu l l matrix vers ion of

this so lve r

is

u s e d

to so lve

nonl inea r equations that arise from our

sequential

modular


49/148

36

simulator. For the sparse

set

of

nonlinear equations in equation-based simulation, the

sparse

vers ion of the modi f ied P owe l l ' s d og l e g method with either

Schubert's update

or

Bogle ' s update is

suitable for solving these equations. Both

vers ions

of

the update

are

implemented.

An

example illustrating the use of our equation so lve r i s

shown

be low.

The

text that appears after

'//'

s a comment statement in C++ M o r e exam

p les are

g i ve n

b y Varma [67] . In i t ia l ly , an array of 1 2 e l e me n t s of Equation-variable

i s

de f ined. These

are the unknowns

in

our equations. Simi la r ly ,

an

array of Equation

is d e c l a r e d so that equations can

be

d e f ine d . F ina l l y , an Equation.set i s n e e d e d

to

store al l

def ined equations.

After

setting

initial

guesses to

the unknowns , the set

of

equations can

be

so lved

eas i ly with

the

'solve()' operation.

In the d e ve lop me n t of both the

full

matrix

and

sparse matrix solvers , w e found

that objects l ike

Matrix, Sparse-matrix, Permutation-vector

and DiagonaLvector are

v e r y helpfu l . A s

an

example, for the Broyden's update, a se c t i on of the code that

performs this

update is

shown in Figure 3.3 .

The

statement

m i m i cs

exactly

the

mathematical

form of

Broyden's

update,

equation

3.2 ,

except

the

last

'()'

in

the

se c ond

statement. In

contrast, one

has to ca l l a

subroutine

o r

inser t

'DO'

l oop s in

a Fortran program to ac c omp l i sh the same

task. Then,

one

l oses

the mathematical

form of

the

or ig ina l equation.

In an

object oriented programming language,

one def ines new data types and

al l

operations that

are deemed log ica l

and

necessary in

manipulating

these ne wly

def ined types. In other wor d s ,

one

can extend the language i t se l f to ac c omp l i sh one's

task.

In most c onve n t iona l programming languages l ike Fortran, data types

of rea l

and

integer

come with the

language.

In addition,

al l

me an ing fu l and

fundamental

mathematical operations l ike '+', '/'?

others

that

one

can perform

on


50/148

37

#include Equation_set.h

#include

bool.h

mainO

c

int

n

= 12; // Order of equation

set

Vector x( eqn2.dat ); // Load initial guesses vector

Equation.variable X[12]; //

Declare number

of unknowns

Equation

e[l2];

// Declare number

of

unknowns

II Define equations using unknown variables defined above

e[0] = 100.0*0.220

-

X[10]*X[0]

-

XCll]*X[5];

eCl]

=

100.0*0.660

-

X[l0]*X[l]

-

X[11]*X[6];

eC2] = 100.0*0.114

-

X[10]*X[2]

-

X[ll]*X[7]

e[3] =

100.0*0.002

-

X[10]*X[3]

- X[ll]*X[8];

e[4] =

100.0*0.004

-

X[10]*X[4]

-

X[ll]*X[9]:

e[5]

=

X[0]

+

XCl]

+

X[2] +

XC3] +

X[4]

- 1.0;

e[6] = X[5] + X[6] +

X[7]

+ X[8] +

X[9]

- 1.0;

e[7] = X[0]

-

66.67*XC5]

e[8] = X[l] -

50.0*X[6]

e[9] = X[2] -

0.015*XC7]

e[10] =

X[3]

-

100.0*X[8]

eCll] = X[4]

-

33.33*X[9]

// Set initial guesses of unknowns

for

(i

= 0; i < n; i++)

X[i]

7/26/2019 Development of a process simulator using object

Development of a process simulator using object oriented programm.pdf

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